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Integrate 1/cos(x)

  1. Sep 14, 2009 #1
    How can you intgertate the 1/cos(x)?

    The right answer can be found in Wolfram alpha at http://www.wolframalpha.com/input/?i=1/cos(x)

    My first wrong answer was ln(cos(x)).
    It suggests me that you cannot use the rule, ln(x), for trigonometric functions
     
  2. jcsd
  3. Sep 14, 2009 #2
    There are several ways to integrate 1/cosx, or secx; just look on Google. You can try
    [tex]\frac{1}{cosx}*\frac{cosx}{cosx} = \frac{cosx}{cos^2x} = \frac{cosx}{1-sin^2x}[/tex]
    and use u = sinx. Or working with secx, use the "clever substitution," as my calc book says, u = secx + tanx, du = sec2x + secxtanx dx. Then substitute u into the previous equation, get u and du together, and integrate.
     
  4. Sep 14, 2009 #3
    Hi soopo
    you can integrata as follows:
    sec x + tan x
    ∫ (1/cos x) dx=∫ sec x dx=∫ sec x ______________ dx
    sec x + tan x

    (sec x)^2 + sec x tan x
    ∫ ___________________ dx
    sec x + tan x
    = ln(sec x + tan x) + C since the numerator is the derivative of the denominator.
    Best Regards
    Riad Zaidan
     
  5. Sep 14, 2009 #4
    Thanks Bohrok!

    I use this

    cosx / (1 - (sinx)^2)

    I get

    1 / (1-u) du = ln|1-u|

    Then, putting u=sinx back to the equation

    1 / (1 - sinx) + C

    ---

    This answer seems to differ from the answer in Wolfram Alpha.
     
  6. Sep 14, 2009 #5

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    No, you get 1/(1- u^2) du

     
  7. Sep 14, 2009 #6
    This may be worth remembering - the general substitution for integrating a rational function of sin, cos R(sin(x),cos(x)), which always works, is:

    - t=cos(x) if R(-u,v)=-R(u,v)
    - t=sin(x) if R(u,-v)=-R(u,v)
    - t=tg(x) if R(-u,-v)=R(u,v)
    - t=tg(x/2) in general
     
  8. Sep 14, 2009 #7
    Thanks for the correction.

    I get

    I [ 1 / (1 - u^2) = .5 ln (1+sinx) - .5 ln (1 - sinx) + C
     
  9. Sep 14, 2009 #8
    1. What is tg?

    2. What is R(-u, v) = -R(u,v)?
     
  10. Sep 14, 2009 #9
    tg = tan

    R(u,v) is the rational function into which you plug sin(x) and cos(x) respectively. If R is "odd with respect to sin", you substitute for cos, and vice versa. t=tan(x/2) is the general substitution which always works (but can be rather cumbersome).
     
  11. Sep 14, 2009 #10

    drizzle

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    Gold Member

    edit, never mind :)
     
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